Ann Oper Res (2010) 178: 5–21DOI 10.1007/s10479-009-0547-y

Stochastic dynamic nursing service budgeting

Gergely Mincsovics · Nico Dellaert

Published online: 20 May 2009© The Author(s) 2009. This article is published with open access at Springerlink.com

Abstract We address the nursing service budgeting problem from the department man-ager’s point of view. The model allocates the budget dynamically to three types of nursingcare capacities: 1) permanent nurses, 2) temporary nurses, and 3) overtime. The quarterlytactical decisions are the aggregate weekly shift pattern of permanent nurses and the policyfor hiring temporary nurses and using overtime. The decisions are optimized with respect tonursing care shortage and a soft-constraint on the annual budget. For the aggregate weeklyshift pattern, permanent nurses require a notification lead-time of one quarter to prepare thepersonal rosters. Our model offers a solution to the nursing service budgeting problem thatextends the existing literature by using a Markovian demand model, resolving the anticipa-tion of the operational decisions, and applying general budget as well as shortage penaltyfunctions.

Keywords Manpower planning · Non-linear stochastic dynamic programming · Healthservice · Optimization · Stochastic processes

1 Introduction

For many hospitals, the costs and availability of nurses are of great concern. How to bestallocate the nursing budget is a complicated problem that the operations research (OR) lit-erature can help address. Today, the existing OR-literature on nurse planning is mainly con-centrated on nurse scheduling models (see e.g. Burke et al. 2004). Interestingly, the nursingservice budgeting problem has been utterly ignored in the last two decades.

The nurse workforce management process comprises decisions that are situated at dif-ferent levels of the decision-making hierarchy. In general three stages are distinguished:

G. Mincsovics (�) · N. DellaertDepartment of Technology Management, Eindhoven University of Technology, P.O. Box 513,5600MB Eindhoven, The Netherlandse-mail: G.Z.Mincsovics@tue.nl

N. Dellaerte-mail: N.P.Dellaert@tue.nl

6 Ann Oper Res (2010) 178: 5–21

budgeting, scheduling, and daily staffing (Brusco and Showalter 1993). It is a complex taskto model the entire hierarchical structure; even the simplest sensible budgeting calculationsare difficult. Cavouras and McKinley (1997) warn for the necessity of feedback reportingamong the decision-makers at different levels. Since the levels of the decision-making hier-archy are all interrelated, isolated optimization of the different levels can easily yield poorperformance (Lowerre 1979). Easton et al. (1992) also point out the importance of integrat-ing staffing and scheduling decisions over a year-long horizon. Following Abernathy et al.(1973), very recently, Li et al. (2007) recall the importance of integrating the decision lev-els in the workforce planning. These two papers suggest iterative methods for solving thehierarchical workforce planning models, they propose.

The more practice-oriented nursing service budgeting literature takes an accountingpoint-of-view to be applied by managers. Kirby and Wiczai (1985) recommend startingwith a basic budgeting system. The simple calculations illustrated in their paper have twosteps: (1) NHPPD (nursing hour per patient day) times annual patient days gives the an-nual workload, (2) annual workload times productivity factor per full-time equivalent hours(contractual FTE) gives the number of full-time nurses to be hired. Arthur and James (1994)review the major workload measurement practices. The exact determination of the produc-tivity factor is an important element of this line of research (Lowerre 1979).

To the best of our knowledge, only two papers contain annual nursing service budgetingmodels in the traditional OR-literature. The primary concern of Trivedi (1981) is to ensurebalanced staffing, to meet union demands, and to satisfy cost control and containment regu-lations, such that the number of part-time and the number of full-time nurses satisfy integrity.This complex problem is modeled as mixed-integer goal programming, which allows em-ploying only deterministic demand. Kao and Queyranne (1985) study a set of models alongthree dimensions: having multi- or single-period, being disaggregate or aggregate for skillclasses, and modeling probabilistic or deterministic demand pattern. We refer the reader toVenkataraman and Brusco (1996) for references to simulation based approaches on inte-grated budgeting and scheduling in services.

The traditional OR-literature does not address important concerns of other, health careoriginated papers. For example, Jeang (1996) alloys the work of Trivedi (1981) and Kao andQueyranne (1985) in order to build a stochastic model that provides the weekly pattern ofthe permanent nurses. The main decision is on the weekly pattern of the permanent nursesas in Trivedi (1981), nevertheless the model accounts for uncertainty of demand, as Kao andQueyranne (1985).

In our paper, similarly to Trivedi (1981), Kao and Queyranne (1985) and Jeang (1996),we introduce a new annual nursing service budgeting model with some additional complex-ities, and illustrate the model’s performance via calculations with real-life data. We alsofollow the mentioned three budgeting papers in not comparing with any previously estab-lished model, but claiming a better representation of real-life. It is the model’s differentoutputs and input needs that impede any of such comparison.

The two major additional complexities that we add are the non-stationary stochasticevolving demand and forecast updates. These aspects are not present in any of the abovementioned papers. Our non-stationary stochastic evolving demand representation allows cal-culation with multiple future demand scenarios at the same time as well as an autocorrelateddemand process. Without forecast updates, capacity decisions are assumed to be made suchthat they do not react to the actual demand realization. In contrast, we use a Markovian de-mand model that allows us to represent forecast updates and our dynamic budget allocationto become responsive to these updates.

We aim at building a model that satisfies the main concerns of health care specialists andnursing managers and meets the modeling requirements of the state of the art in OR. To

Ann Oper Res (2010) 178: 5–21 7

achieve this goal, we use the principles of OR literature in health care for conceptual mod-eling, the guidelines of health care specialists for modeling capacity decisions, and nursingmanagers’ accounting suggestions for determination of important cost aspects. This paperthus helps the communication between OR specialists and health care managers as well asserves as a good basis for developing practical nursing service budgeting models.

In the following two sections, we introduce a set of descriptive models. These are theshortage penalty cost, the budget penalty, the demand and the productivity models. In theend of Sect. 3, we propose a decision structure and build a conceptual optimization model forthe stochastic dynamic nurse budgeting problem. Simplifications of this conceptual modelresult in the final computational model, which is introduced in Sect. 4. We demonstrate theusefulness of the computational model via performing numerical experiments in Sect. 5.Conclusions are drawn in Sect. 6.

2 Longitudinal service budget allocation

This section presents a simple optimization model, which introduces the general notionof the longitudinal service budget allocation. The model represents the budget allocationprocess focusing on the trade-off between service capacity shortage and budget deviation.The severity of different levels of service capacity shortages and budget deviations are de-scribed by penalty functions, which we discuss in more details.

2.1 Capacity shortage penalty function

The concept of shortage penalty cost (a virtual cost) shows a general way of modeling lossof quality due to capacity shortage. This concept was developed for the case of nursingservices by Warner and Prawda (1972). It is more general than the service level concept, seee.g. Jeang (1996), who restricts capacity allocation.

According to the shortage penalty concept, predefined penalty costs are assigned to ca-pacity shortages. The following axiomatic statements characterize the shortage penalty func-tion: (1) the penalty is positive in case of shortage and zero if there is no shortage; and (2) thecost is convexly (and non-linearly) increasing function of the shortage.

In our model, we use a generalized shortage penalty cost model. We assume that theshortage penalty cost is an arbitrary time-dependent function of the demand for nursing careand the available nursing capacity in the period.

2.2 Budget deviation penalty function

Although some existing models use penalty functions for budget deviations, the concept ofsuch penalty has not been axiomatically described yet. We mention that the penalty functionin the nursing service budgeting model of Trivedi (1981) is assumed to be linear, and that themore abstract service budgeting model of Zimmerman (1976) uses a general function form.In our model, we also use a general form of budget penalty function, which can depend onthe annual budget and the annual capacity costs.

2.3 Conceptual optimization model for longitudinal budget allocation

We present a conceptual optimization model, (OM1), to demonstrate the use of shortage andbudget penalty costs. The given budget W is distributed over periods with the objective to

8 Ann Oper Res (2010) 178: 5–21

find a balance between shortages and budget deficit.

Model 1—Conceptual static optimization model for the budgeting problem (OM1)

mincapacity

{(∑t

ShortagePenalty(demandt, capacityt)

)

+ BudgetPenalty(W,Cost(capacity))

}.

The budget W is thus a resource, which is to be optimally allocated for employing capacityto minimize the shortages and budget deviations. Naturally, the shortage and budget penaltyfunctions need to have the same unit of utility, so that we can calculate a trade-off betweenthem. Our further models preserve the trade-off idea of this conceptual model, and extend itby a detailed decision structure and demand dynamics.

3 Stochastic dynamic nursing service budgeting

Many details are missing from OM1 that are specific to the budgeting of nursing service.Since our goal is to build a nursing service budgeting model, we need to gain understandinghow the demand patterns look, what the productive part of the service capacity is, as well asthe capacity sources and capacity decisions are.

3.1 Demand pattern characteristics

In order to find the most appropriate demand model, we need to gain understanding ofits characteristics. Warner and Prawda (1972) find that demand prediction for nursing carehas 5–10% error for a few days ahead. However, after the first few days, error increasedto 20–30%. The relatively small error in the short-term allow them to use a deterministicdemand model with a twice a week rolling schedule. Kao and Queyranne (1985) comparetime-dependent stochastic demand models having independent, monthly periods with theirdeterministic counterparts. Their results indicate that ignoring demand uncertainty can leadto an underestimate of budget needs. By the computations, Kao and Queyranne (1985) usean ARIMA model to generate a stochastic demand model with independent periods. Kaoand Tung (1980) show that the number of monthly patient days at different departmentsfit different ARIMA models. The type of their best fit ARIMA models suggest that theassumption of independent demand periods is not appropriate. Generally, the assumption ofindependent periods is not justified and unwanted.

In our model, we use a general stochastic demand process model with a general de-pendency structure. The only assumption is that the demand process is exogenous, i.e., ourdecisions do not influence the demand. Similarly to Kao and Queyranne (1985), we simplifythe demand model for our computations. However, next to the time-dependency of demand,we allow dependencies between demands of different periods.

3.2 Productivity

The calculation of productivity gives a substantial part of the nursing capacity cost account-ing. Keeling (1999) explains that we can expect to have 1,477 hours productive work (pro-ductive FTE) from the 2,080 hours contracted (contractual FTE). Lowerre (1979) lists hol-idays, vacations, personal days, and sick days that constitute the non-productive fraction ofthe contractual FTE, and shows a sequential procedure for accurate calculation.

Ann Oper Res (2010) 178: 5–21 9

Table 1 Some descriptive information on nursing from Siferd and Benton (1992)

Starting time (on weekdays) three 49%, four to five 29%

Shift pattern have most or all staff with a permanent shift assignment 49%

have most or all staff who work a set pattern of days on and off 33%

have most or all staff who work the same days each week 21%

Overtime authorize nursing overtime 100 to 400 times per year 42%

always use voluntary overtime for a shortage of nursing staff 54%

Temporary nurses authorize use of temporary nursing staff 30 or more times per year 43%

sometimes use hospital pool nurses for a shortage of nursing staff 62%

Hiring (and leaves) hire new nursing staff 2 to 9 times per year 61%

One can find productivity constants in models as either the ratio of productive and con-tractual FTE as p in Kao and Queyranne (1985), or its reciprocal, as γ in Trivedi (1981). Weemploy the former, ‘p’ productivity definition in our model. Contrary to the deterministicproductivity formulations in the literature, we take sample paths of productivity to describeits random behavior shift by shift (Pti). To evaluate the available (productive) workforce ina shift, which is supposed to be an integer, we use controlled stochastic rounding (R(.)).

3.3 Capacity decisions

In this section, we make our modeling choices for the decisions, their timing and structure,and extend the optimization model OM1 to a more detailed one, OM2. After a short reviewof the axiomatic models, we recall some results of an empirical study that can serve as agood basis for the model extension.

Practical computational models on budgeting are concerned with calculating the numberof full-time, part-time nurses on pay-roll, the overtime to be utilized, and a usually weeklong pattern that repeats throughout the budgeted horizon (Trivedi 1981; Jeang 1996), ormake decisions only at an aggregate, monthly level (Kao and Queyranne 1985). All thesemodels fail to represent the dynamics, if the capacity decisions are taken responsively to therecent demand realizations and the actual remaining budget. Consequently, these models donot anticipate future capacity response (to e.g. a sustaining low level of demand), and do notbenefit from the fact that most of the capacity decisions need not to be made in the beginningof the horizon. In Sect. 3.1, we mentioned that the demand forecast error can increase from5–10% for a few days ahead to 20–30% on a longer term. This observation suggests thatdelaying the decision making must have some added value.

A better source of input for modeling the decision structure is the empirical literature.Siferd and Benton (1992) surveys hospital nursing units providing useful statistics on nurs-ing service capacity decisions. These statistics are summarized in Table 1 outlining howoften certain capacity options are used among their respondents. Afterwards, we group thecapacity decisions by their frequency, creating a new table, Table 2. Here, we include thedecisions’ lead-time in brackets, additionally. We deduce the hierarchical structure of deci-sions based on this table to build the model, OM2.

The starting time abbreviates the number of shifts and the daily time slots that the shiftscover. This is a decision at the strategic level: the feasible set of shift timings remains un-changed for years. In our model, we assume three shifts each day with some exogenous,fixed starting times.

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Table 2 Decisions (and their lead times) grouped by their frequency

years starting time (0)

yearly/quarterly shift pattern (1), hiring and leaves (1)

per shift temporary nursing (0 or more), overtime (0 or more)

The shift pattern stands for the aggregate weekly pattern of permanent nurses. That is,21 values describing the regular number of permanent nurses in each shift of the week. Al-ternatively, the shift pattern can be for example two-week long. Our model calculates theaggregate shift patterns as well as the hiring decisions, which may be necessary, (approx-imately) quarterly. Naturally, it is more practical to shorten or prolong the quarters so thatthose start and end at the time of leaves or in times of better hiring opportunities.

We note that the determination of the number of permanent nurses from their aggregateshift pattern is simple: the total of the aggregate shift pattern needs to be multiplied by theratio of the single shift workload per year, 365 × 8 h = 2920 h, and the productive FTE.

The operational level consists of overtime and temporary nurse hiring decisions. Weassume a limited overtime, enough supply to reach the overtime limit, and an infinite supplyof temporary nurses that we can unlimitedly use, as in Kao and Queyranne (1985). In ourmodel, we assume to limit the overtime according to a predefined policy, which declares aset of feasible overtime values (Z). Temporary nurse hiring is mostly an ad-hoc operationaldecision. It is a separated short-term decision even in the best practical models (Bard andPurnomo 2006), not reckoning with the budget constraints on the long term. In the nextsubsection, we propose a model that overcomes the coordination problem of the operationaland higher level decisions.

3.4 Conceptual optimization model for nursing service budgeting

We formulate a conceptual stochastic dynamic programming model for the budgeting prob-lem, OM2, which uses the concept of OM1 and the descriptive models. The model, OM2includes the demand and productivity aspects as explained in Sects. 3.1 and 3.2, and uses thedecision structure of Table 2. In the model, the tactical and operational decisions create anembedded structure of (penalty) cost-to-go functions: the f (.) functions correspond to thetactical decisions, and the g(.) functions to the operational ones. The quarterly cost-to-gof (.) has t index, while the shift index of g(.) is i. There are T = 5 quarters since we includethe first quarter’s permanent shift pattern decision made one quarter in advance before thebudgeted year starts (OM2.B); quarter t has It shifts.

In line with OM1, the only costs are the budget penalty and some shortage penalties(see sti (. , .) in (OM2.D) and B(. , .) in (OM2.G)). The model’s state space consists ofthe remaining budget, the demand state, and the shift pattern for the next quarter. By theoperational decisions, we need the current quarter’s shift pattern, additionally. E.[.] standsfor the expected value operator.

Model 2—Conceptual stochastic dynamic nurse budgeting optimization model (OM2)

f0 = f1(W,m1), (OM2.A)

f1(r1,m1) = minu2

{ED1(m1)[f2(r2, u2,D(D1(m1)))]} with r2 = r1, (OM2.B)

ft (rt , ut ,mt ) = minut+1

{EDt (mt ),Pt [gt,1(rt − cuut , ut , ut+1, Dt (mt ))]}, (OM2.C)

Ann Oper Res (2010) 178: 5–21 11

gt,i(rt,1, ut , ut+1, Dt (mt )) = minvti ,oti

{sti(Dt i (mt ),R(uti , Pt i ) + vti + oti) + gt,i+1(rt,1

− vticvj − otic

oj , ut , ut+1, Dt (mt ))}, (OM2.D)

gt,It +1(rt,It +1, ut , ut+1, Dt (mt )) = ft+1(rt,It +1, ut+1,D(Dt (mt ))), (OM2.E)

fT (rT , uT ,mT ) = EDT (mT ),PT[gT,1(rT − cuuT , uT ,0, Dt (mt ))], (OM2.F)

fT +1(W, rT +1) = B(W,W − rT +1). (OM2.G)

The tactical decisions can be seen in (OM2.B, OM2.C, OM2.F, OM2.G), which give thequarterly dynamics of the model, whereas the operational decisions are modeled in (OM2.D,OM2.E), which give the dynamics at the shift level. Equation (OM2.A) defines the minimalexpected annual quality loss for a budget, W , and an initial demand state, m1. Equation(OM2.B) determines the first nurse shift pattern of the budgeted year, u2. Equation (OM2.C)decides on the permanent shift pattern of a quarter ahead, where the feasible shift pattern, ut ,is a 21 element long non-negative integer vector. Equation (OM2.D) describes the decisionsper shift, where we minimize penalty costs for a given demand by making the best choicefor overtime and temporary nurses. The feasible overtime values, vti are from the set, Z.The number of temporary nurses can be any non-negative integer value. Equation (OM2.E)calculates the transition to the new demand state, based upon the demand pattern realizationsthat correspond to the previous demand state, mt . Equation (OM2.F) is similar to (OM2.C),but as this is the last quarter no shift pattern decision is made any more. Finally, in (OM2.F),takes into consideration the consequences of end-of-year budget deviations, how much thestarting budget has been depleted by the costs of permanent nurses (OM2.C, OM2.F), andthe costs of temporary help (OM2.D).

We represent the evolving demand by a set of year-long sample paths. These samplepaths are categorized into groups in each quarter (e.g., low/medium/high total demand inthe quarter). The groups are associated with demand states (mt). The quarterly demand statetransition probabilities can be calculated as the number of paths changing demand stateaccordingly.

The demand state serves as the tactical level information source that let model the dy-namics of forecast updates. The tactical level decision, ut , is based on the knowledge of theremaining budget and the present demand state, from which the set of relevant path contin-uations can be extracted. For each of the relevant paths, a deterministic problem is solvedresulting in the operational level decisions, vti and oti .

Table 3 summarizes the notations of variables used. The second column declares thevariable category: input (I), output (O), and auxiliary (–). In the latter category, we classifiedoutputs of no particular importance.

4 Computational model

Since OM2 has more than thousand decision epochs and a large state space, it is not attrac-tive computationally. We build another model, which is a simplified version of OM2, and assuch, its minimizations can be evaluated. We take two simplification steps.

4.1 Computational optimization model for the stochastic dynamic nursing servicebudgeting problem

In the first simplifying step, we reduce the number of decision epochs to the number ofperiods plus one by creating a hierarchical optimization structure. The numerous decision

12 Ann Oper Res (2010) 178: 5–21

Table 3 Notations (variable, variable category (input I, output O, and auxiliary –), description)

indices

t – period index (t = 1, . . . ,5, quarters; quarters 2, . . . ,5 are budgeted)i – subperiod index within a period (i = 1, . . . , It ,8h shifts, where It = 270,273 or 276)j (t, i) – (we abbreviate it to j ) subperiod index within a midperiod (j = 1, . . . ,21,8h shifts of

the week)

capacity decisions

ut O = (u1, . . . , u21) aggregate weekly shift pattern of permanent nurses, where uj is thenumber of permanent nurses in each weekly subperiod j of the actual period, andsimilarly, u1 is the number of permanent nurses in subperiod i of the actual period

Vt O temporary help policy; it gives our preferred exchange rate between the penalty costsand the real money

vi O the number of temporary pool nurses hired in subperiod i of the actual period(vi = 0,1,2, . . .)

oi O the amount of overtime utilized in subperiod i of the actual period (oi ∈ Z)

capacity restrictions

Z I the set of feasible amount of overtime per shift

cost functions

ft (.) O expected minimal penalty-cost-to-go from the beginning of period t onwardsgt,i (.) O expected minimal penalty-cost-to-go from the beginning of subperiod i of period t

onwards

cost coefficients

cuj

I unit shiftly cost of a permanent nurse in the subperiod j of a week

cvj

I unit shiftly cost of a temporary pool nurse in the subperiod j of a week

coj

I unit shiftly cost of overtime in the subperiod j of a week

budget and cost

W I annual budgetrt – remaining budget in the beginning of period t

C(d, u,V ) O capacity cost subtotal in the actual period for demand sample path d, aggregatepermanent midperiodly shift pattern ii, and temporary help policy V

penalty cost functions

St (d, u,V ) O nursing care shortage penalty cost subtotal in period t for demand sample path d,aggregate permanent weekly shift pattern u, and temporary help policy V

sj (d, c) I nursing care shortage penalty cost in the subperiod j of a week, when demand fornursing care is d, and the nursing capacity is c

B(W,C) I budget penalty cost for annual budget W and for annual capacity costs C

demand

mt – demand state that captures information about the demand process’ of period t in thebeginning of the period

Dt (mt ) I Markovian demand process in period t

di – demand realization in subperiod i of the actual periodD(d) – demand state transition function, e.g. if the demand realization in period t was d , it

gives the next demand state mt+1; it can be interpreted as forecast information forperiod t + 1

Ann Oper Res (2010) 178: 5–21 13

Table 3 (Continued)

productivity

Pt i I a random process that gives the fraction of the productive permanent nursing capacity

R(x) – ={

�x� with probability x − �x��x� with probability �x� − x

, a random variable that helps generate randomized

integers from the real number x

epochs of OM2 are a result of the operational decisions. Our goal is to find tactical decisionsthat define the operational decisions. Therefore, we assume that the operational decisionsfollow a policy, which is decided at the tactical level. We call this policy the temporary helppolicy Vt , renewed each quarter. By this way the operational decisions form a consistent partof the budget allocation. For example, if we expect to end up with budget deficit, we will beless willing to use overtime or hire temporary nurses.

In the second simplifying step, we reduce the state space via a mapping. It is the aggregatepermanent shift pattern that makes the state space large. We carry 21 dimensions of the shiftpattern next to few other dimensions. We propose an approximation that helps in resolvingthe curse of dimensionality for this situation. Namely, we create a mapping ut (mt−1,B

ut ),

which calculates the aggregate shift pattern from only two dimensions: the demand stateat the time of the decision mt−1, and the budget part allocated to cover the shift patternof permanent nurses Bu

t . This way the multiple dimensions of ut are translated to the twodimensions mt−1 and Bu

t . The necessary mapping we calculate via a greedy algorithm.The apparent simplicity compared to OM2 is a result of the disappearing optimizations

(OM2.D) and (OM2.E) and the reduced state space dimensionality. Equation (SDNBOM.A)represents the minimal expected annual penalty costs for a budget W and an initial demandstate (m0,m1). Equation (SDNBOM.B) determines the first quarter’s permanent nurse bud-get, while (SDNBOM.C) also decides on the temporary help policy parameter, Vt , minimiz-ing the expected future penalty costs for the remaining budget.

Model 3—Stochastic dynamic nurse budgeting optimization model (SDNBOM)

f0 = f1(W,m0,m1) (SDNBOM.A)

f1(r1,m0,m1) = minBu

2

{ED1 [f2(r2,m1,D(D1),Bu2 )]} with r2 = r1 (SDNBOM.B)

ft (rt ,mt−1,mt ,But ) = min

Vt ,But+1

{EDt ,Pt [St (Dt , ut , Vt )

+ ft+1(rt − cuut − C(Dt , ut , Vt ),mt ,D(Dt ),But+1)]}

(SDNBOM.C)

fT (rT ,mtT −1,mT ,BuT ) = min

VT

{EDT ,PT[ST (DT , uT ,VT )

+ fT t+1(W, rT − cuuT − C(DT , uT ,VT )))]} (SDNBOM.D)

fT +1(W, rT +1) = B(W,W − rT +1) (SDNBOM.E)

where ut and Dt abbreviates ut (mt−1,But ) and Dt (mt−1,mt ), respectively.

A fortunate advantage of SDNBOM is that the state space transformation with theut (mt−1,B

ut ) mapping imported a new dimension, mt−1, to the state space of period t . This

14 Ann Oper Res (2010) 178: 5–21

implies that we can afford to use a second-order Markovian demand model. We remarkthat some extra calculations are necessary: in (SDNBOM.C) and (SDNBOM.D) we need toevaluate the St (.), the C(.) functions, and ut .

4.2 Algorithms for the simplifying calculations

In this section, we discuss implementation issues of algorithms that can evaluate the St (.),and C(.) functions, and the ut vectors. The algorithms, we present are examples and notintended to provide the optimal operational decisions. Instead, the provided operational de-cisions are reasonable and, importantly, coordinated with the tactical decisions.

Algorithm 1 calculates the shortage penalty and capacity cost St (.) and C(.) for a giventemporary help policy and sample path, optimizing the temporary help and overtime. Thetemporary help policy class has a single parameter, Vt , which gives our preferred exchangerate between the quality-related penalty costs and the money allocated from the budgetfor temporary help, in quarter t . We invest into overtime and/or temporary nurses up tothe capacity level where the shortage penalty/capacity cost ratio in the shift get closest toVt while not exceeding it. The policy parameter Vt , we optimize in the beginning of eachquarter. For the calculations, we take a number of realizations of Dt as a function of mt−1,which are sample paths (vectors) with elements di , the demand in shift i.

Algorithm 2 is a greedy algorithm, which calculates the aggregate permanent shift pat-tern, ut . Here, Qj(k) = EDt [

∑the weekly index of

shift i inquarter t is jsi(Dti , k)] is defined as the expected sum of

shortage penalties in the actual period incurred if k nursing capacity is used for the weeklyindex j (e.g., j is ‘Monday night’, then Qj(5) is the sum of shortage penalties of Mondaynight shifts throughout the period if 5 permanent nurses are hired). (Qj(0))j is the vector ofQj(0)’s having elements for all j ’s. Bu is a given upper bound for Bu

t . �Q is the vector ofthe actual Qj gradients. The algorithm provides the table (Bu

i , ui) for all shift i, for somegiven quarter and demand state.

Algorithm 1 Calculation of shortage penalty and capacity costs St (.) and C(.)

qi(v, o) = si(di,R(ui Pt i ) + v + o)

uicui + vcv

i + ocoi

(vi, oi) = arg max(v,o)∈N0×Z

qi (v,o)≤Vt

qi(v, o)

C(d,ut ,Vt ) =∑

i

vcvi + oco

i

St (d,ut ,Vt ) =∑

i

si(di,R(ui Pt i ) + vi + oi)

4.3 Justification of the simplifying steps

The policy class that Algorithm 1 represents is an approximation, which ignores the demand-and budget-responsiveness of the temporary help decisions within the quarter, but it re-mains quarterly responsive. Under the assumption that the quarters are static (the demand

Ann Oper Res (2010) 178: 5–21 15

Algorithm 2 Calculation of the permanent shift pattern ut (mt−1,But )

(i,Bui , ui,Q) := (0,0,0, (Qj (0))

j)

�Q :=(

Qj − Qj(1)

cuj

)j

while Bui+1 ≤ Bu

j∗ := arg maxj

{�Qj }

ui+1 := ui

(Bui+1, u

i+1j∗ ) := (Bu

i + cuj∗, u

i+1j∗ + 1)

Qnew := Qj∗(ui+1j∗ )

(�Qj∗,Qj∗) := (Qj∗ − Qnew,Qnew)

i := i + 1

end while

and budget circumstances do not change), Algorithm 1 can provide optimal temporary helpdecisions. Namely, if si(di, .) is convex for all i and all possible demand value di , then Al-gorithm 1 becomes equivalent with a greedy algorithm yielding an optimal behavior (Fox1966). In the greedy algorithm, Vt becomes the terminating gradient value. Since Vt is opti-mized, the budget spent for temporary help in the quarter is also optimized. Note that if theyear is split into more periods, then the approximation improves.

The optimality of Algorithm 2’s greedy mechanism may be damaged by poor anticipationof the future use of temporary help. We may assume that either no temporary help is used orwe use temporary help only as replacement in case of absenteeism of the permanent workers(Warner and Prawda 1972). Under any of these assumptions, the greedy Algorithm 2 givesan optimal solution (Fox 1966).

4.4 Discussion of the assumptions

Since the nursing service budgeting is a composite problem, many assumptions need to betaken while modeling. We create an explicit list of our modeling assumptions so that one canjudge the models applicability. The assumptions are ordered by their time span decreasingly.

Assumption 1 Three shifts a day meaning three fixed starting times.

Siferd and Benton (1992) reports that 49% of the surveyed hospitals use three startingtimes. In the cases, where more starting times are in use, appropriate alternative algorithmsto Algorithm 1 and 2 are difficult to find. Provided these algorithms, however, our model isstill applicable without taking this assumption.

Assumption 2 Demand for nursing care is purely exogenous, independent from the capac-ity level, the quality of care provided or other controllable variables.

16 Ann Oper Res (2010) 178: 5–21

Demand for care is not independent on the capacity levels, in general. On the one hand,when staffing at adequate levels patients’ stays are shorter than by consistently short staffing,because then adequate nursing care is received (Flood and Diers 1988). On the other hand, bya continuously unsatisfactory level of nursing capacity, patients may tend to select anotherhospital because of the low quality or the long waiting times. Our assumption can, however,get justification since our model optimizes capacity decisions for a given budget. The budgetlimits the long-term level of nursing capacity that can be provided, so both the capacity andthe service level can be, approximately, regarded as constant.

Assumption 3 Years are independent; no long-term effects are taken into account.

On the strategic long-term, the surrounding population of patients and nurses can increaseor decrease. Costs of capacities can change; new regulations may come into effect. Wedo not model these aspects, although the annual change in the patient population can beincorporated into the demand model.

Assumption 4 In the end of each quarter nurses can be hired or fired in unlimited amountat no cost.

This assumption can be restrictive in times of a nursing shortage (Brusco and Showalter1993). For the situations, where hiring cannot be solved easily, we suggest using a modified,constrained version of SDNBOM, where we constrain the search space of the aggregate shiftpattern of permanent nurses (e.g., |∑j ut+1,j − ∑

j ut,j | ≤ 5 for some t).

Assumption 5 Once we set up an aggregate weekly shift schedule, new permanent nursesare hired, and the permanent nurses establish a set of personal rosters to meet the aggregateschedule in one quarter.

We can use period lengths different from a quarter, which are suitable to describe thelead-time of hiring and personal roster negotiations. Note that our model allows variableperiod lengths as well.

Assumption 6 Budget is known a quarter in advance, before the budgeted year starts.

For the case, if the budget is not known by the time, we decide on the coming year’sfirst shift pattern, we can say, it is known in stochastic terms. Again, we suggest usingan altered version of SDNBOM: an expectation on the budget may follow the first stage’sminimization.

Assumption 7 Nursing care shortage depends only on demand, capacity and time.

Although this type of shortage penalty function is a broad generalization of that in Warnerand Prawda (1972), it can still carry restrictions to particular situations. For example, a fur-ther generalization to dependency on demand, permanent capacity, overtime, temporary ca-pacity and time may be preferable. The SDNBOM model allows this generalization withoutany additional calculational complexity.

Assumption 8 Budget penalty depends only on the given annual budget and the annualcosts.

Ann Oper Res (2010) 178: 5–21 17

We generalized the linear soft budget constraint of Trivedi (1981) to an arbitrary penaltyfunction. We are not aware of any sensible broader generalization.

Assumption 9 No difference in efficiency between nurses.

This assumption relates to Assumption 7: we can translate the possible efficiency differ-ences to differences in the generalized shortage penalty function.

Assumption 10 Demand forecast error is negligible in the short-term: demand during ashift is assumed to be known at the beginning of the shift.

Naturally, this assumption can be seriously restrictive, when a large fraction of incom-ing patients is emergency type, and the claimed closely deterministic short-term demandstructure is not valid. Otherwise, it is a reasonable assumption (Warner and Prawda 1972).

Assumption 11 Temporary nurses can be hired only for whole shifts.

Assumption 12 Overtime for less than one shift.

If we needed more nursing care that the permanent nurses can provide, we will usetemporary nurses or overtime. As long as we can calculate the best feasible overtime—temporary nurse combination from the desired temporary help capacity, we can modify theSDNBOM model to cover any temporary capacity policy till the shifts are independent. I.e.,the SDNBOM model cannot treat policies that have constraints on a set of shifts, e.g., if onehad overtime last weekend, she would not be allowed to have overtime this weekend.

Assumption 13 Temporary nurses and overtime volunteers have infinite supply; we canhire them in the beginning of the shift.

Temporary nurses and overtime volunteers are not generally always available (Bruscoand Showalter 1993). Constant upper limits on their number can be included in our model.Alternatively, by gradually increasing the hourly cost of temporary nurse hiring, we can alsolower the use of temporary nurses to some given limit.

Assumption 14 No carry-over of workload from shift to shift (lost service).

In the call-center staffing literature, Atlason et al. (2005) point out that service of con-secutive (short) time periods are interrelated, demand is partially lost, and partially carriedover to the next period. We can expect the same interrelation to hold for demand for nursingcare.

Assumption 15 Single nurse class and substitution between classes.

If there are fixed ratios between nursing classes, the permanent capacity cost will beapproximately linear function of the permanent capacity. For the better application of thegreedy algorithm, we need to have this permanent capacity cost function being convexlyincreasing (Fox 1966). We do not handle substitution between classes.

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Table 4 The pure and the relative quadratic penalty cost functions

pure quadratic relative quadratic

ShortagePenalty = (demand − capacity)2 ShortagePenalty = ( demand−capacitydemand

)2

5 Numerical experiments

In this section, we first demonstrate what kind of solution SDNBOM can provide. After-wards, we show experiments investigating the shortage penalty cost model selection.

To illustrate the functioning of SDNBOM, we used workload data of a mental healthinpatient ward (Ridley 2007). The maximum of nursing care demand per shift was eightnurses; the daily total demand was about 11, with around 6, 3, and 1 in the day, evening, andnight shifts, respectively. The number of demand sample paths was twelve, which we gainedby simulating the ARMA demand process fit to the workload data. The more sample paths,the better the demand process is represented, and the more memory and calculation time isneeded. The number of productivity sample paths was three, with an average productivityaround 70%. The number of demand states per period was three. Under this setting, oneevaluation of the SDNBOM took around 5 minutes (main computer parameters: 2.8 GHzCPU, 1 Gb RAM).

We modeled the capacity shortage costs as being dependent both on the demand andthe available capacity, resulting in the relative quadratic penalty cost function (see Table 4),which we considered more realistic than the pure quadratic shortage penalty cost functionof Warner and Prawda (1972). Namely, as opposed to the relative quadratic penalty, the purequadratic penalty has a shortcoming in that it regards the situation with one demand and nonurses as severe as having ten demand and nine nurses.

While solving SDNBOM with the setting described above, we archived the state-dependent decisions throughout the year. This archive allowed us to show for the differentdemand sample paths how the budget is allocated longitudinally (see Fig. 1), and how farcapacity is matched to demand (Fig. 2).

In Fig. 1, we can see how the budget is allocated to permanent capacity and temporarycapacities (including overtime) along the year for 12 demand scenarios. Each quarter startswith a step downwards, which corresponds to the quarterly permanent capacity expendi-tures. Within each quarter, some part of the budget is consumed by the temporary capacityexpenses. By the end of the year, the remaining budget finishes around zero. For some sce-narios, this could only be reached by allowing considerable capacity shortage costs in thelast quarter.

Figure 2 depicts the match between demand and capacity for each shift and each scenario,quarterly grouped. Although the majority of the points are on the shortage side because ofthe limited budget, SDNBOM provides a good match between demand and capacity.

In our further experiments, we evaluated the SDNBOM with pure and relative quadraticpenalty cost functions and tested the impact of modeling updated forecasts. We compare theoutcomes in Table 5. The table demonstrates the solution of the first quarter’s permanentshift pattern for an annual budget of 8,000 and forecast updates. Because of the random-ization in the SDNBOM, we can get different results for the same parameter setting. Therepeated experiments under the same setting showed that using the pure quadratic penaltythe SDNBOM sometimes assigns zero nurses to night shifts (still overtime and temporarylabor can be used) and more fluctuations in the number of permanent nurses in a shift ofthe week. To the contrary, using the relative quadratic penalty seems to result in overstaffingnight shifts. Although the SDNBOM does not enable us to justify the use of any penalty

Ann Oper Res (2010) 178: 5–21 19

Fig. 1 Illustration of theremaining budget in the course ofthe year for 12 demand scenarios

Fig. 2 Demand vs. the available capacity in the four quarters for the same 12 demand scenarios as in Fig. 1.The diagonal line indicates from where shortage penalty is to be paid (below the line)

Table 5 Outcomes and outcomes ranges under the same parameter settings for the pure and the relativequadratic penalty cost functions based on 25 runs

Pure quadratic Permanent shift pattern for the first quarter (ut ) Total FTE

Bu2 D E N D E N D E N D E N D E N D E N D E N

median 2,038.3 9 5 2 9 5 2 9 5 2 9 5 2 9 5 2 9 5 2 9 5 2 22.0

minimum 1,428.3 7 2 0 7 3 0 7 3 0 7 3 0 7 3 0 7 2 0 7 3 0 15.0

maximum 2,074.7 10 5 2 9 5 2 10 5 2 10 5 2 9 5 2 10 5 2 9 5 2 22.4

Relative quadratic Permanent shift pattern for the first quarter (ut ) Total FTE

Bu2 D E N D E N D E N D E N D E N D E N D E N

median 1,561.7 5 4 3 5 4 3 6 4 3 5 4 3 6 4 3 5 4 3 5 4 3 17.4

minimum 1,433.9 4 4 3 4 4 3 5 4 3 4 4 3 5 4 3 5 4 3 4 4 3 16.2

maximum 1,861.5 7 5 3 7 5 3 7 5 3 7 5 3 7 4 3 7 5 3 7 4 3 20.6

cost functions, we can conclude that the results are sensitive on the selection of the shortagepenalty cost model, and that the preferences of the hospital management could play a largerole in deciding on the penalty cost function.

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In our further comparisons, we evaluated the SDNBOM with and without modeling fore-cast updates. In this example, we found that the penalty cost reduction gained from usingquarterly forecast updates is 91%. However, for some stable demand process, we generated,the cost reduction becomes marginal, between 0–2%. We note that the response to the quar-terly forecast updates is sometimes a big change in the permanent nursing capacity, whichis not necessarily wanted.

6 Conclusions

We built a stochastic dynamic optimization model for the nursing service budgeting prob-lem based on existing concepts. We generalized some of these concepts, and in few caseswe gave suggestions for further generalizations. First, we built a conceptual optimizationmodel, which consisted of exact descriptive models of the generalized concepts. As far asthe underlying concepts and the data are reliable, the OM2 conceptual model provided op-timal decisions. Due to its high complexity, we could not evaluate the conceptual model.Therefore, we proposed some simplifying steps, which led to our final optimization model,SDNBOM. We verified the results the SDNBOM provide and illustrated how far it makesdemand and capacity match. The SDNBOM model, we could evaluate in some minutes ona single personal computer, for multiple periods.

By building our model, we put emphasis on the precise modeling of reality. Althoughwe cannot expect the results to be optimal, we used our model to test assumptions on theshortage penalty cost models and on the demand forecast updates. We found that differ-ent shortage penalty cost functions can lead to quite different staffing decisions. Therefore,the management should carefully select a shortage penalty cost function that appropriatelyrepresents their preferences. All in all, our overview and our findings can supplement thedevelopment of future practical computational models on nursing service budgeting.

Future research, may address empirical modeling of the shortage and budget penaltyfunctions. Using empirical penalty functions would make further numerical experimentswith the SDNBOM interesting. Additionally, studying a set of real-life demand processeswould be necessary to draw appropriate conclusions on the value of using forecast updates.Furthermore, it would be interesting to formulate a mixed integer program instead of thesimple greedy approximation of Algorithm 2.

Acknowledgements Herewith we thank to Colin Ridley for sending the data that he collected in Ridley(2007) and for providing us with many additional details. We also thank Marion Rauner, and the anonymreviewers for their suggestions towards the improvement of this paper.

Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommer-cial License which permits any noncommercial use, distribution, and reproduction in any medium, providedthe original author(s) and source are credited.

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